Fusion Ring
\[ \begin{array}{llllllll} \htmlTitle{1\otimes 1}{1} & & & & & & & \\ \htmlTitle{2\otimes 1}{2} & \htmlTitle{2\otimes 2}{1} & & & & & & \\ \htmlTitle{3\otimes 1}{3} & \htmlTitle{3\otimes 2}{4} & \htmlTitle{3\otimes 3}{1 \oplus 5} & & & & & \\ \htmlTitle{4\otimes 1}{4} & \htmlTitle{4\otimes 2}{3} & \htmlTitle{4\otimes 3}{6 \oplus 2} & \htmlTitle{4\otimes 4}{1 \oplus 5} & & & & \\ \htmlTitle{5\otimes 1}{5} & \htmlTitle{5\otimes 2}{6} & \htmlTitle{5\otimes 3}{3 \oplus 7} & \htmlTitle{5\otimes 4}{8 \oplus 4} & \htmlTitle{5\otimes 5}{1 \oplus 5 \oplus 8} & & & \\ \htmlTitle{6\otimes 1}{6} & \htmlTitle{6\otimes 2}{5} & \htmlTitle{6\otimes 3}{8 \oplus 4} & \htmlTitle{6\otimes 4}{3 \oplus 7} & \htmlTitle{6\otimes 5}{7 \oplus 6 \oplus 2} & \htmlTitle{6\otimes 6}{1 \oplus 5 \oplus 8} & & \\ \htmlTitle{7\otimes 1}{7} & \htmlTitle{7\otimes 2}{8} & \htmlTitle{7\otimes 3}{5 \oplus 8} & \htmlTitle{7\otimes 4}{7 \oplus 6} & \htmlTitle{7\otimes 5}{3 \oplus 7 \oplus 6} & \htmlTitle{7\otimes 6}{5 \oplus 8 \oplus 4} & \htmlTitle{7\otimes 7}{1 \oplus 5 \oplus 8 \oplus 4} & \\ \htmlTitle{8\otimes 1}{8} & \htmlTitle{8\otimes 2}{7} & \htmlTitle{8\otimes 3}{7 \oplus 6} & \htmlTitle{8\otimes 4}{5 \oplus 8} & \htmlTitle{8\otimes 5}{5 \oplus 8 \oplus 4} & \htmlTitle{8\otimes 6}{3 \oplus 7 \oplus 6} & \htmlTitle{8\otimes 7}{3 \oplus 7 \oplus 6 \oplus 2} & \htmlTitle{8\otimes 8}{1 \oplus 5 \oplus 8 \oplus 4} \\ \end{array} \]
Frobenius-Perron Dimensions
| Simple | Numeric | Symbolic |
|---|---|---|
| \( 1\) | \(1.000\) | \( 1 \) |
| \( 2\) | \(1.000\) | \( 1 \) |
| \( 3\) | \(1.879\) | \( \cos{\left(\frac{4 \pi}{9} \right)} + \cos{\left(\frac{2 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} \) |
| \( 4\) | \(1.879\) | \( \cos{\left(\frac{4 \pi}{9} \right)} + \cos{\left(\frac{2 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} \) |
| \( 5\) | \(2.532\) | \( - \cos{\left(\frac{4 \pi}{9} \right)} + \cos{\left(\frac{2 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} + 1 \) |
| \( 6\) | \(2.532\) | \( - \cos{\left(\frac{4 \pi}{9} \right)} + \cos{\left(\frac{2 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} + 1 \) |
| \( 7\) | \(2.879\) | \( \cos{\left(\frac{4 \pi}{9} \right)} + \cos{\left(\frac{2 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} + 1 \) |
| \( 8\) | \(2.879\) | \( \cos{\left(\frac{4 \pi}{9} \right)} + \cos{\left(\frac{2 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} + 1 \) |
| \( D^2\) | 38.469 | \(12 \cos{\left(\frac{2 \pi}{9} \right)} + 12 \cos{\left(\frac{\pi}{9} \right)} + 18\) |
Modular Data
Twist Factors
\[ \begin{pmatrix} \htmlTitle{\theta_{1}}{0} & \htmlTitle{\theta_{2}}{\frac{3}{2}} & \htmlTitle{\theta_{3}}{\frac{1}{6}} & \htmlTitle{\theta_{4}}{\frac{2}{3}} & \htmlTitle{\theta_{5}}{\frac{4}{9}} & \htmlTitle{\theta_{6}}{\frac{35}{18}} & \htmlTitle{\theta_{7}}{\frac{5}{6}} & \htmlTitle{\theta_{8}}{\frac{4}{3}} \end{pmatrix} \]
S Matrix
\[ \left(\begin{array}{llllllll} \htmlTitle{S_{1; 1}}{1} & & & & & & & \\ \htmlTitle{S_{2; 1}}{1} & \htmlTitle{S_{2; 2}}{-1} & & & & & & \\ \htmlTitle{S_{3; 1}}{-\zeta_{72}^{20} + \zeta_{72}^{8} + \zeta_{72}^{4}} & \htmlTitle{S_{3; 2}}{\zeta_{72}^{20} - \zeta_{72}^{8} - \zeta_{72}^{4}} & \htmlTitle{S_{3; 3}}{-\zeta_{72}^{20} + \zeta_{72}^{8} + \zeta_{72}^{4} + 1} & & & & & \\ \htmlTitle{S_{4; 1}}{-\zeta_{72}^{20} + \zeta_{72}^{8} + \zeta_{72}^{4}} & \htmlTitle{S_{4; 2}}{-\zeta_{72}^{20} + \zeta_{72}^{8} + \zeta_{72}^{4}} & \htmlTitle{S_{4; 3}}{\zeta_{72}^{20} - \zeta_{72}^{8} - \zeta_{72}^{4} - 1} & \htmlTitle{S_{4; 4}}{\zeta_{72}^{20} - \zeta_{72}^{8} - \zeta_{72}^{4} - 1} & & & & \\ \htmlTitle{S_{5; 1}}{-\zeta_{72}^{16} + \zeta_{72}^{8} + \zeta_{72}^{4} + 1} & \htmlTitle{S_{5; 2}}{-\zeta_{72}^{16} + \zeta_{72}^{8} + \zeta_{72}^{4} + 1} & \htmlTitle{S_{5; 3}}{-\zeta_{72}^{16} + \zeta_{72}^{8} + \zeta_{72}^{4} + 1} & \htmlTitle{S_{5; 4}}{-\zeta_{72}^{16} + \zeta_{72}^{8} + \zeta_{72}^{4} + 1} & \htmlTitle{S_{5; 5}}{0} & & & \\ \htmlTitle{S_{6; 1}}{-\zeta_{72}^{16} + \zeta_{72}^{8} + \zeta_{72}^{4} + 1} & \htmlTitle{S_{6; 2}}{\zeta_{72}^{16} - \zeta_{72}^{8} - \zeta_{72}^{4} - 1} & \htmlTitle{S_{6; 3}}{\zeta_{72}^{16} - \zeta_{72}^{8} - \zeta_{72}^{4} - 1} & \htmlTitle{S_{6; 4}}{-\zeta_{72}^{16} + \zeta_{72}^{8} + \zeta_{72}^{4} + 1} & \htmlTitle{S_{6; 5}}{0} & \htmlTitle{S_{6; 6}}{0} & & \\ \htmlTitle{S_{7; 1}}{-\zeta_{72}^{20} + \zeta_{72}^{8} + \zeta_{72}^{4} + 1} & \htmlTitle{S_{7; 2}}{\zeta_{72}^{20} - \zeta_{72}^{8} - \zeta_{72}^{4} - 1} & \htmlTitle{S_{7; 3}}{1} & \htmlTitle{S_{7; 4}}{-1} & \htmlTitle{S_{7; 5}}{\zeta_{72}^{16} - \zeta_{72}^{8} - \zeta_{72}^{4} - 1} & \htmlTitle{S_{7; 6}}{-\zeta_{72}^{16} + \zeta_{72}^{8} + \zeta_{72}^{4} + 1} & \htmlTitle{S_{7; 7}}{\zeta_{72}^{20} - \zeta_{72}^{8} - \zeta_{72}^{4}} & \\ \htmlTitle{S_{8; 1}}{-\zeta_{72}^{20} + \zeta_{72}^{8} + \zeta_{72}^{4} + 1} & \htmlTitle{S_{8; 2}}{-\zeta_{72}^{20} + \zeta_{72}^{8} + \zeta_{72}^{4} + 1} & \htmlTitle{S_{8; 3}}{-1} & \htmlTitle{S_{8; 4}}{-1} & \htmlTitle{S_{8; 5}}{\zeta_{72}^{16} - \zeta_{72}^{8} - \zeta_{72}^{4} - 1} & \htmlTitle{S_{8; 6}}{\zeta_{72}^{16} - \zeta_{72}^{8} - \zeta_{72}^{4} - 1} & \htmlTitle{S_{8; 7}}{-\zeta_{72}^{20} + \zeta_{72}^{8} + \zeta_{72}^{4}} & \htmlTitle{S_{8; 8}}{-\zeta_{72}^{20} + \zeta_{72}^{8} + \zeta_{72}^{4}}\end{array}\right) \]
Central Charge
\[c = \frac{7}{3} \]